Propagation Dynamics Of A Class Of Population Models For Signle Species In A Discrete Periodic Habitat

Propagation of invasive species in heterogeneous habitat is a popular topic.How to characterize the influence of environmental heterogeneity on propagation dynamics is an interesting but challenging question,since different heterogeneous habitats may have very distinct influences on the dynamics.This thesis is concerned with one dimensional periodic lattice habitat,assuming that odd locations are bad for species while even locals are good.With this biological scenario,we propose a stage-structured population model in heterogeneous habitat.Then we study its dynamics under different situations.Firstly,we analyze the influence of dispersal rate on the propagation speed when the model admits the monostable structure.We obtain the fundamental solution to a linear system with countably many ODEs,as well as its continuity with respect to initial values under the compact open topology.With these properties,we are able to apply monotone dynamical system theory on wave propagation to obtain the existence of spreading speed and its coincidence with the minimal speed of traveling waves.Then by using the eigenvalues for related problems we have the variational characterization of the spreading speed.With the dispersal rate from even location to odd location as the parameter,we prove the existence of the optimal dispersal rate to maximize the speed.Finally,we obtain an explicit characterization for both the optimal dispersal rate and the corresponding maximal speed.Secondly,we analyze the influence of the dispersal rate on propagation success or failure when the model admits the bistbale structure.Introducing the Allee effect yields a bistable structure.Then we apply the monotone dynamical system theory to obtain the existence of bistable traveling waves.By appealing to the squeezing technique we prove the uniqueness of wave speed and profile,as well as the exponential stability of bistable waves up to translations.With the dispersal rate from even location to odd location as the parameter,we found a threshold value for propagation failure or success.Moreover,we found there is an optimal dispersal rate to maximize the speed when propagation succeeds.Lastly,we study the global dynamics when the growth is of unimodal type and the initial values are periodic in space.Unimodal growth make the model does not admit the comparison principle in general.With restricted initial values,the model is reduced to a system of two equations.Then we employ the exponential ordering,fluctuation method and the eigenvalue analysis to study the global attractivity of steady state by using time delay as the parameter.